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If we study $p$-adic Galois representations we come across the so called "Robba rings", defined as the ring of Laurent series which converge in some annulus $\{z| \,\,R<|z|<1\}$ for some positive number $R<1$. I looked at the literature a bit to see why such rings are important, but I could not find any that explains the genesis of the concept itself. Apparently these rings are intimately associated to $(\Phi,\Gamma)$- modules. Any reference or any hint why Robba rings are studied? Perhaps the motivation is from the usual complex analysis?

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    $\begingroup$ I knew a little about this some years ago, and will try to remember and write up an answer. Besides, I think this question would be appropriate for MathOverflow, as this is a hot research topic, and there are certainly more competent people than me who hopefully would give an answer. $\endgroup$ – Torsten Schoeneberg Oct 18 '18 at 3:44

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